Noah Hanifin scored to make it 2-0 at 13:51 of the second period. Swayman made the save on a shot by Tkachuk, but the rebound went right to Hanifin to the left of the crease.
"For me individually, it's always fun coming back home and playing Boston," Hanifin said. "… To get the win and play the way we did as a team was great. Obviously coming into a tough building like Boston, it's huge to get the first two goals. That helped us out a little bit."
Mangiapane scored shorthanded at 3:08 of the third period to give the Flames a 3-0 lead. Swayman made two saves against Dillon Dube before Mangiapane put in the rebound after Dube's third attempt hit the side of the net.
Mikael Backlund made it 4-0 at 4:18 with a shot from the top of the right circle that trickled between Swayman's pads. Bruins forward Charlie Coyle tried to pull the puck off the goal line, but video review showed the puck crossed the line.
Calgary was coming off a 5-2 win at the New York Islanders on Saturday, when Boston won 5-2 at the Philadelphia Flyers.
NOTES: Vladar is the first Flames goalie to shut out the Bruins since Miikka Kiprusoff on Dec. 17, 2005. … Mangiapane has scored five goals in the past three games and is the first Flames player to score 15 goals through their first 19 games in a season since Jarome Iginla in 2001-02. … Fourteen of Mangiapane's goals have come on the road. He is the third player in NHL history to score at least 14 of his first 15 goals of a season on the road (John LeClair, 14 in 1994-95 with Montreal Canadiens and Philadelphia; Paul Gardner, 14 in 1976-77 with Colorado Rockies). … Calgary scored the first goal for the NHL-high 15th game this season, two more than the next closest (Washington Capitals, 13 games). … Flames forward Elias Lindholm played his 600th NHL game. He is the second player from the 2013 NHL Draft to reach the mark, joining teammate Sean Monahan (610 games).